To find the vertices of the square inscribed in the circle given by the equation \(x^2 + y^2 - 2x + 4y - 93 = 0\), we will follow these steps:
### Step 1: Rewrite the Circle Equation
We start by rewriting the circle equation in standard form. The general form of a circle is:
\[
x^2 + y^2 + 2gx + 2fy + c = 0
\]
From the given equation, we can identify:
- \(2g = -2\) → \(g = -1\)
- \(2f = 4\) → \(f = 2\)
- \(c = -93\)
### Step 2: Find the Center and Radius
The center \((h, k)\) of the circle can be found using:
\[
(h, k) = (-g, -f) = (1, -2)
\]
Next, we calculate the radius \(r\) using the formula:
\[
r = \sqrt{g^2 + f^2 - c}
\]
Substituting the values:
\[
r = \sqrt{(-1)^2 + (2)^2 - (-93)} = \sqrt{1 + 4 + 93} = \sqrt{98} = 7\sqrt{2}
\]
### Step 3: Calculate the Diameter
The diameter \(D\) of the circle is twice the radius:
\[
D = 2r = 2 \times 7\sqrt{2} = 14\sqrt{2}
\]
### Step 4: Relate the Square's Side to the Circle's Diameter
For a square inscribed in a circle, the diagonal of the square equals the diameter of the circle. If \(a\) is the side length of the square, then the diagonal \(d\) is given by:
\[
d = a\sqrt{2}
\]
Setting this equal to the diameter:
\[
a\sqrt{2} = 14\sqrt{2}
\]
Dividing both sides by \(\sqrt{2}\):
\[
a = 14
\]
### Step 5: Determine the Vertices of the Square
Since the square is inscribed in the circle and its sides are parallel to the coordinate axes, the vertices of the square can be determined by moving \(7\) units (half of the side length) from the center in both the x and y directions.
1. **Center**: \((1, -2)\)
2. **Vertices**:
- Vertex A: Move left and up → \((1 - 7, -2 + 7) = (-6, 5)\)
- Vertex B: Move right and up → \((1 + 7, -2 + 7) = (8, 5)\)
- Vertex C: Move right and down → \((1 + 7, -2 - 7) = (8, -9)\)
- Vertex D: Move left and down → \((1 - 7, -2 - 7) = (-6, -9)\)
### Final Vertices
Thus, the vertices of the square are:
- A: \((-6, 5)\)
- B: \((8, 5)\)
- C: \((8, -9)\)
- D: \((-6, -9)\)
### Answer
The vertices of the square are \((-6, 5)\), \((8, 5)\), \((8, -9)\), and \((-6, -9)\).
---
